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1、定义
运载体的俯仰角、横滚角和航向角统称为姿态角
运载体的姿态角是根据运载体坐标系 相对地理坐标系的转角确定的
2、航向角 ψ\psiψ
1、定义：地理坐标系 yty_tyt​轴与载体坐标系横滚轴在水平面的投影的yhy_hy..." />
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1、定义
运载体的俯仰角、横滚角和航向角统称为姿态角
运载体的姿态角是根据运载体坐标系 相对地理坐标系的转角确定的
2、航向角 ψ\psiψ
1、定义：地理坐标系 yty_tyt​轴与载体坐标系横滚轴在水平面的投影的yhy_hy...">
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1、定义
运载体的俯仰角、横滚角和航向角统称为姿态角
运载体的姿态角是根据运载体坐标系 相对地理坐标系的转角确定的
2、航向角 ψ\psiψ
1、定义：地理坐标系 yty_tyt​轴与载体坐标系横滚轴在水平面的投影的yhy_hy...">
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                                    2.2 2.3 姿态角 方向余弦法和欧拉角
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                                    2023-09-14, 592 words, 3 min read
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                                        <h2 id="一-姿态角">一、姿态角</h2>
<h4 id="1-定义">1、定义</h4>
<p>运载体的俯仰角、横滚角和航向角统称为姿态角</p>
<p>运载体的姿态角是根据运载体坐标系 相对地理坐标系的转角确定的</p>
<h4 id="2-航向角-psi">2、航向角 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">ψ</span></span></span></span></h4>
<p>1、定义：地理坐标系 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">y_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>轴与载体坐标系横滚轴在水平面的投影的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>y</mi><mi>h</mi></msub></mrow><annotation encoding="application/x-tex">y_h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>轴的夹角，偏东为正</p>
<p>0~360</p>
<img src="http://cos.pansis.site/202309142331919.png" alt="image-20230914233140864" style="zoom:33%;" />
<h4 id="3-俯仰角-theta">3、俯仰角 $\theta $</h4>
<p>1、定义：：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>y</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">y_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>轴与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>y</mi><mi>h</mi></msub></mrow><annotation encoding="application/x-tex">y_h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>轴的夹角， 抬头为正（高于水平面为正）</p>
<p>-90~90</p>
<h4 id="4-横滚角-gamma">4、横滚角 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span></span></span></span></h4>
<p>1、定义：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">x_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>轴与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>h</mi></msub></mrow><annotation encoding="application/x-tex">x_h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>轴的夹角， 右倾为正， 左倾为负。</p>
<p>-180~180</p>
<h2 id="二-方向余弦法">二、方向余弦法</h2>
<h4 id="1-矢量的方向余弦">1、矢量的方向余弦</h4>
<p>1、定义：设取直角坐标系<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>o</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">oxyz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">o</span><span class="mord mathdefault">x</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span></span></span></span> ，沿坐标轴的单位矢量分别为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span> 、<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span> ，并设过原点有一矢量 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span> ，它在各坐标轴上的坐标分别为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">R_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">R_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">R_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<img src="http://cos.pansis.site/202309142345713.png" alt="image-20230914234549665" style="zoom:33%;" />
<p>投影 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">R_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">R_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">R_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>可表示为<img src="http://cos.pansis.site/202309142347601.png" alt="image-20230914234718546" style="zoom: 67%;" /></p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span>表示矢量<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span>的模长。</p>
<p><img src="http://cos.pansis.site/202309142348059.png" alt="image-20230914234822018" style="zoom: 50%;" />称为矢量R的方向余弦。</p>
<h4 id="2-坐标系变换方向余弦矩阵">2、坐标系变换（方向余弦矩阵）</h4>
<p>1、方法：方向余弦矩阵</p>
<p>2、同一矢量<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span>从坐标系<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0,y_0,z_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>变换为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>z</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1,y_1,z_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<figure data-type="image" tabindex="1"><img src="http://cos.pansis.site/202309221536683.png/abc123" alt="image-20230922153635636" loading="lazy"></figure>
<p>其中，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mo>(</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">cos()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">s</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span>代表两坐标系之间的夹角方向余弦。</p>
<p>3、方向余弦矩阵</p>
<p>$C_{0}^{r} $ 称为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> 系对 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span> 系的方向余弦矩阵，如上图所示。</p>
<p>$C_{r}^{0} $ 称为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span> 系对 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> 系的方向余弦矩阵</p>
<img src="http://cos.pansis.site/202309221542049.png/abc123" alt="image-20230922154246994" style="zoom: 33%;" />
<p>4、方向余弦矩阵的性质</p>
<img src="http://cos.pansis.site/202309221543090.png/abc123" alt="image-20230922154320943" style="zoom:33%;" />
<h4 id="3-例题">3、例题</h4>
<img src="http://cos.pansis.site/202309221547263.png/abc123" alt="image-20230922154637771" style="zoom:33%;" />
<img src="http://cos.pansis.site/202309221547420.png/abc123" alt="image-20230922154653808" style="zoom:33%;" />
<h2 id="三-欧拉角法">三、欧拉角法</h2>
<h4 id="1-原理">1、原理</h4>
<p>刚体坐标系相对于参考坐标系的角位置，可以用三次独立的转角来确定。</p>
<p>对于定点转动的刚体，只要给定 <strong>一组欧拉角</strong> ，就能唯一确定刚体坐标系的九个方向余弦，从而唯一地确定刚体 在空间的角位置。</p>
<h4 id="2-欧拉角">2、欧拉角</h4>
<p>这三次独立的旋转角度。</p>
<p>欧拉角的选取不是唯一的，不同的旋转顺序会有不同的欧拉角。</p>
<p>➢ 第一次转动可以 <strong>绕刚体坐标系</strong> 的任意一根轴进行；</p>
<p>➢ 第二次转动可以 <strong>绕其余两根轴中</strong> 的任意一根轴进行；</p>
<p>➢ 而第三次转动可以 <strong>绕第二次转动以外的两根轴中的</strong>  任意一根轴进行。</p>
<h4 id="3-欧拉角下的方向余弦矩阵">3、欧拉角下的方向余弦矩阵</h4>
<img src="http://cos.pansis.site/202309221551422.png/abc123" alt="image-20230922155156282" style="zoom:50%;" />
<img src="http://cos.pansis.site/202309221554879.png/abc123" alt="image-20230922155434789" style="zoom:33%;" />
<img src="http://cos.pansis.site/202309221556328.png/abc123" alt="image-20230922155629222" style="zoom: 33%;" />
<img src="http://cos.pansis.site/202309221556306.png/abc123" alt="image-20230922155643202" style="zoom: 33%;" />
<img src="http://cos.pansis.site/202309221557056.png/abc123" alt="image-20230922155718014" style="zoom:25%;" />
<figure data-type="image" tabindex="2"><img src="http://cos.pansis.site/202309221557057.png/abc123" alt="image-20230922155729991" loading="lazy"></figure>
<h4 id="4-例题">4、例题</h4>
<figure data-type="image" tabindex="3"><img src="http://cos.pansis.site/202309221603156.png/abc123" alt="image-20230922160346986" loading="lazy"></figure>
<img src="http://cos.pansis.site/202309221603706.png/abc123" alt="image-20230922160355596" style="zoom:33%;" />
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